Average Error: 1.5 → 1.5
Time: 35.1s
Precision: 64
\[\frac{\left(\left(-b\right) - \left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)\right)}{\left(\left(2\right) \cdot a\right)}\]
↓
\[\frac{1.0}{2} \cdot \frac{\left(-b\right) - \sqrt{\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), 4, \left(a \cdot c\right)\right)\right)}}{a}\]
\frac{\left(\left(-b\right) - \left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)\right)}{\left(\left(2\right) \cdot a\right)}
↓
\frac{1.0}{2} \cdot \frac{\left(-b\right) - \sqrt{\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), 4, \left(a \cdot c\right)\right)\right)}}{a}
double f(double a, double b, double c) {
        double r444517 = b;
        double r444518 = -r444517;
        double r444519 = r444517 * r444517;
        double r444520 = 4.0;
        double r444521 = /* ERROR: no posit support in C */;
        double r444522 = a;
        double r444523 = c;
        double r444524 = r444522 * r444523;
        double r444525 = r444521 * r444524;
        double r444526 = r444519 - r444525;
        double r444527 = sqrt(r444526);
        double r444528 = r444518 - r444527;
        double r444529 = 2.0;
        double r444530 = /* ERROR: no posit support in C */;
        double r444531 = r444530 * r444522;
        double r444532 = r444528 / r444531;
        return r444532;
}
↓
double f(double a, double b, double c) {
        double r444533 = 1.0;
        double r444534 = 2.0;
        double r444535 = r444533 / r444534;
        double r444536 = b;
        double r444537 = -r444536;
        double r444538 = r444536 * r444536;
        double r444539 = /*Error: no posit support in C */;
        double r444540 = 4.0;
        double r444541 = a;
        double r444542 = c;
        double r444543 = r444541 * r444542;
        double r444544 = /*Error: no posit support in C */;
        double r444545 = /*Error: no posit support in C */;
        double r444546 = sqrt(r444545);
        double r444547 = r444537 - r444546;
        double r444548 = r444547 / r444541;
        double r444549 = r444535 * r444548;
        return r444549;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 1.5

    \[\frac{\left(\left(-b\right) - \left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)\right)}{\left(\left(2\right) \cdot a\right)}\]
  2. Using strategy rm

  3. Applied introduce-quire1.5

    \[\leadsto \frac{\left(\left(-b\right) - \left(\sqrt{\left(\color{blue}{\left(\left(\left(b \cdot b\right)\right)\right)} - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)\right)}{\left(\left(2\right) \cdot a\right)}\]

  4. Applied insert-quire-fdp-sub1.5

    \[\leadsto \frac{\left(\left(-b\right) - \left(\sqrt{\color{blue}{\left(\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), \left(4\right), \left(a \cdot c\right)\right)\right)\right)}}\right)\right)}{\left(\left(2\right) \cdot a\right)}\]
  5. Using strategy rm

  6. Applied p16-*-un-lft-identity1.5

    \[\leadsto \frac{\left(\left(-b\right) - \color{blue}{\left(\left(1.0\right) \cdot \left(\sqrt{\left(\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), \left(4\right), \left(a \cdot c\right)\right)\right)\right)}\right)\right)}\right)}{\left(\left(2\right) \cdot a\right)}\]

  7. Applied p16-*-un-lft-identity1.5

    \[\leadsto \frac{\left(\color{blue}{\left(\left(1.0\right) \cdot \left(-b\right)\right)} - \left(\left(1.0\right) \cdot \left(\sqrt{\left(\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), \left(4\right), \left(a \cdot c\right)\right)\right)\right)}\right)\right)\right)}{\left(\left(2\right) \cdot a\right)}\]

  8. Applied distribute-lft-out--1.5

    \[\leadsto \frac{\color{blue}{\left(\left(1.0\right) \cdot \left(\left(-b\right) - \left(\sqrt{\left(\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), \left(4\right), \left(a \cdot c\right)\right)\right)\right)}\right)\right)\right)}}{\left(\left(2\right) \cdot a\right)}\]

  9. Applied p16-times-frac1.5

    \[\leadsto \color{blue}{\left(\frac{\left(1.0\right)}{\left(2\right)}\right) \cdot \left(\frac{\left(\left(-b\right) - \left(\sqrt{\left(\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), \left(4\right), \left(a \cdot c\right)\right)\right)\right)}\right)\right)}{a}\right)}\]

  10. Final simplification1.5

    \[\leadsto \frac{1.0}{2} \cdot \frac{\left(-b\right) - \sqrt{\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), 4, \left(a \cdot c\right)\right)\right)}}{a}\]

Reproduce

herbie shell --seed 2019153 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  (/.p16 (-.p16 (neg.p16 b) (sqrt.p16 (-.p16 (*.p16 b b) (*.p16 (real->posit16 4) (*.p16 a c))))) (*.p16 (real->posit16 2) a)))